Analysis

A27

(∫_(-1)^1)(∫_(-(Sqrt[1 - (x^2)]))^((Sqrt[1 - (x^2)])))1 \[DifferentialD]y \[DifferentialD]x

inneres Integral

int = (∫_(-(Sqrt[1 - (x^2)]))^((Sqrt[1 - (x^2)])))1 \[DifferentialD]y

Stammfunktion

stamm = ∫int \[DifferentialD]x

A 28

(∫_(-1)^1)(∫_(-(1 - z))^(1 - z))(∫_(-(Sqrt[(((1 - z))^2) - (x^2)]))^((Sqrt[(((1 - z))^2) - (x^2)])))1 \[DifferentialD]y \[DifferentialD]x \[DifferentialD]z

NIntegrate[
      1, {z, -1, 1}, {x, -(1 - z), 
        1 - z}, {y, -(Sqrt[(((1 - z))^2) - (x^2)]), (Sqrt[(((1 - z))^2) - (x^2)])}]

inneres Integral

int = (∫_(-(Sqrt[(((1 - z))^2) - (x^2)]))^((Sqrt[(((1 - z))^2) - (x^2)])))1 \[DifferentialD]y

nächstes Integral

Stammfunktion

stamm = ∫int \[DifferentialD]x

Simplify[stamm, z > 0]

bestimmtes Integral

int = (∫_(-(1 - z))^(1 - z))(∫_(-(Sqrt[(((1 - z))^2) - (x^2)]))^((Sqrt[(((1 - z))^2) - (x^2)])))1 \[DifferentialD]y \[DifferentialD]x

int = 
      Integrate[(∫_(-(Sqrt[(((1 - z))^2) - (x^2)]))^((Sqrt[(((1 - z))^2) - (x^2)])))1 \[DifferentialD]y, {x, -(1 - z), 1 - z}, 
        Assumptions -> z < 1 && z > 0]

Simplify[int, 0 < z  &&  z < 1]

gesamtes Integral

resultat = (∫_0^1) int \[DifferentialD]z

N[resultat]

A 29

(a)

RotationsVolumen[
        f_, {x_, a_, 
          b_}] := π (∫_a^b) 
                (f^2) \[DifferentialD]x

Kugel: 3dimensional

ParametricPlot3D[{Cos[ϕ] Sin[θ], Sin[ϕ] Sin[θ], 
        Cos[θ]}, {ϕ, 0, 2  π}, {θ, 
        0, π}]

Kugel: 2dimensionale Projektion

Plot[(Sqrt[1 - (x^2)]), {x, -1, 1}, 
      AspectRatio -> Automatic]

Kugel: Volumen

RotationsVolumen[(Sqrt[(r^2) - (x^2)]), {x, -r, r}]

Torus: 3dimensional

ParametricPlot3D[{Cos[θ], (2 + Sin[θ]) 
          Cos[ϕ], (2 + Sin[θ]) Sin[ϕ]}, {ϕ, 0, 
        2  π}, {θ, 0, 2  π}]

Torus: 2dimensionale Projektion

Plot[{2 + (Sqrt[1 - (x^2)]), 2 - (Sqrt[1 - (x^2)])}, {x, -1, 1}, 
      AspectRatio -> Automatic, 
      PlotStyle -> {RGBColor[1, 0, 0], RGBColor[0, 0, 1]}, 
      PlotRange -> {0, 3}]

Torus: Volumen

vol = 
      RotationsVolumen[R + (Sqrt[(r^2) - (x^2)]), {x, -r, r}] - 
        RotationsVolumen[R - (Sqrt[(r^2) - (x^2)]), {x, -r, r}]

Simplify[vol, r > 0]

(c) Länge eines Graphen

GraphLänge[
        f_, {x_, a_, 
          b_}] := (∫_a^b)(Sqrt[ 1 + 
                D[f, x]^2]) \[DifferentialD]x

Beispiel: halber Kreisumfang

GraphLänge[(Sqrt[1 - (x^2)]), {x, -1, 1}]

Lineare Algebra

L26

A = {{0, 1, 0}, {1, 0, 3}, {0, 3, 0}}

charpol = CharacteristicPolynomial[A, λ]

Nach dem Satz von Cayley-Hamilton ist also

MatrixPower[A, 3] == 10*A

und damit für ungerade Exponenten (A^(2  n + 1)) = (10^n) A sowie nach Multiplikation mit A für gerade Exponenten (A^(2  n + 2)) = (10^n) (A^2)

Inverse[A]

L27

A = (1/2) {{3, 0, 1}, {0, -2, 0}, {1, 0, 3}}

CharacteristicPolynomial[A, λ]

sys = Eigensystem[A]

B = Transpose[sys[[2]]]

Inverse[B] . A . B